Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

4.1.4 Distribution factors
4. The collective model: a formal analysis 161
We may now reintroduce distribution factors. An interesting feature is that
such factors do not change the Pareto frontier, but only the Pareto weight.
In geometrical terms, thus, they can only generate moves along the Pareto
frontier (from II to III in Figure 3.8). This suggests that analyzing the
impact of distribution factors may help understanding the nature and the
form of such moves. This intuition can be given a formal translation. Equation
(4.4) above can now be rewritten as:
ˆg (r,x,z) =˜g (r,x,μ(r,x,z)) (4.13)
Because the same μ (.) function appears in all goods the collective model
yields cross-equation restrictions. To see this, consider the consequences of
a marginal change in distribution factor zk on the collective demand for
commodity i:
∂ˆgi
=
∂zk
∂˜gi ∂μ
(4.14)
∂μ ∂zk
Comparing the effect of different distribution factors, say zk and zl, wefind
that (assuming ∂gi/∂zl 6= 0):
∂ˆgi/∂zk
∂ˆgi/∂zl
= ∂μ/∂zk
∂μ/∂zl
(4.15)
The right hand side term is independent of the good we are considering.
Hencewehavetheproportionality property that the ratio of derivatives
with respect to two sharing factors is the same for all goods. The result
that the impact of zk and zl must be proportional across commodities is
very important empirically, and can be given various equivalent forms; for
instance, we can write that 5
∂ˆgi
=
∂zk
∂μ/∂zk
.
∂μ/∂zl
∂ˆgi
(4.16)
∂zl
If the impact of a change in zk on household demand for good i is, say, twice
as large as that of zl, then the same must be true for all commodities; and
we can actually conclude that the impact of zk on the Pareto weight μ is
twice as large as that of zl. Intuitively, whatever the number of distribution
factors, they only operate through their impact on μ; hence their impact
is one-dimensional. In a sense, it is as if there was one distribution factor
only. This prediction is empirically testable (subject to having at least two
distribution factors); possible tests will be discussed in the next chapter.
Another interesting feature of (4.14) is that it provides additional information
about the structure of price and income effects in the collective
5 Equivalently, the matrix Dzg with general terms ∂g i
∂z k is of rank (at most) one.